Related papers: Notes on semisimple tensor categories of rank two

We calculate the number of the isomorphism class of the finite flat models over the ring of integers of an absolutely ramified $p$-adic field of constant group schemes of rank two over finite fields, by counting the rational points of a…

We give explicit formulas for the asymptotic growth rate of the number of summands in tensor powers in certain monoidal categories with finitely many indecomposable objects, and related structures.

We study the monoidal closed category of symmetric multicategories, especially in relation with its cartesian structure and with sequential multicategories (whose arrows are sequences of concurrent arrows in a given category). Then we…

In that paper, we prove that the collection of all FRBSU monoidal categories and the collection of all crossed modules form a 2 category.

We show that finding rank-$R$ decompositions of a 3D tensor, for $R\le 4$, over a fixed finite field can be done in polynomial time. However, if some cells in the tensor are allowed to have arbitrary values, then rank-2 is NP-hard over the…

For a connected semisimple algebraic group $G$, we consider some special infinite series of tensor products of simple $G$-modules whose $G$-fixed point spaces are at most one-dimensional. We prove that their existence is closely related to…

We develop abstract nonsense for module categories over monoidal categories (this is a straightforward categorification of modules over rings). As applications we show that any semisimple monoidal category with finitely many simple objects…

Over an arbitrary field, we prove that the relative 2-Deligne tensor product of two separable module 2-categories over a compact semisimple tensor 2-category exists. This allows us to consider the Morita 4-category of compact semisimple…

For any finite field $\mathbb{F}$ and any positive integer $n$ we count the number of monic polynomials of degree $n$ over $\mathbb{F}$ with nonzero constant coefficient and a self-reciprocal factor of any specified degree. An application…

We consider sets with infinite addition, called $\Sigma$-monoids, and contribute to their literature in three ways. First, our definition subsumes those from previous works and allows us to relate them in terms of adjuctions between their…

Given a semi-simple pre-Tannakian category over a finite field, we show that (a slight modification of) its linearization over a field of characteristic 0 is also semi-simple and pre-Tannakian. The key input is a result of Kuhn on the…

Tannaka duals of Hopf algebras inside semisimple tensor categories are used to construct orbifold tensor categories, which are shown to include the Tannaka dual of the dual Hopf algebras. The second orbifolds are then canonically isomorphic…

We construct free monoids in a monoidal category with finite limits and countable colimits, in which tensoring on either side preserves reflexive coequalizers and colimits of countable chains.

We categorify the notion of an infinitesimal braiding in a linear strict symmetric monoidal category, leading to the notion of a (strict) infinitesimal 2-braiding in a linear symmetric strict monoidal 2-category. We describe the associated…

We introduce analogues of Soergel bimodules for complex reflection groups of rank one. We give an explicit parametrization of the indecomposable objects of the resulting category and give a presentation of its split Grothendieck ring by…

The primary contribution of this paper is to give a formal, categorical treatment to Penrose's abstract tensor notation, in the context of traced symmetric monoidal categories. To do so, we introduce a typed, sum-free version of an abstract…

In this paper we count the number of some particular quadruple persymmetric rank i matrices over F_2.

We start the general structure theory of not necessarily semisimple finite tensor categories, generalizing the results in the semisimple case (i.e. for fusion categories), obtained recently in our joint work with D.Nikshych. In particular,…

We prove a categorical duality between a class of abstract algebras of partial functions and a class of (small) topological categories. The algebras are the isomorphs of collections of partial functions closed under the operations of…

This paper introduces methods for classifying actions of finite-dimensional Hopf algebras on path algebras of quivers, and more generally on tensor algebras $T_B(V)$ where $B$ is semisimple. We work within the broader framework of finite…